What is your motivation for this? And what is your understanding of the situation up until now?
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David RobertsDec 13 '11 at 23:56

My understanding of present knowledge is that the answer is it's not known, but highly doubtful that such an algorithm exists.
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Ryan BudneyDec 14 '11 at 0:40

1

I think current run-time estimates for known algorithms to construct the prime decomposition of a knot tend to be at least exponential. I'm thinking of normal surface theory arguments.
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Ryan BudneyDec 14 '11 at 0:41

6

A straightforward comment is that this problem is at least as hard as recognizing the unknot. (Given a diagram D for knot, form a larger diagram as the "connected sum" of D with a trefoil diagram. The larger diagram represents a composite knot iff D represents the unknot.)
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Peter KronheimerDec 14 '11 at 13:45

I know that in computational topology, there are several softwares like SnapPea and Regina, designed to help topologists in the resolutions of hard problems. I am studying knot theory in this period and then i would like to know if it is possbile (thanks to these programs or other softwares) to decompose a (non prime) knot by an algorithm that runs in polynomial time!
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GianDec 14 '11 at 17:53